EXAMPLES AND APPLICATIONS OF AND K-THEORY

JONATHAN ROSENBERG

Abstract. These are informal notes from my course at the 3era Escuela de Invierno Luis Santal´o-CIMPA Research School on Topics in Noncommutative Geometry. Feedback, especially from participants at the course, is very wel- come. The course basically is divided into two (related) sections. Lectures 1–3 deal with Kasparov’s KK-theory and some of its applications. Lectures 4–5 deal with one of the most fundamental examples in noncommutative geometry, the noncommuative 2-torus.

Contents Lecture 1. Introduction to Kasparov’s KK-theory 3 1.1. Why KK? 3 1.2. Kasparov’s original definition 4 1.3. Connections and the product 7 1.4. Cuntz’s approach 8 1.5. Higson’s approach 9 Lecture 2. K-theory and KK-theoryofcrossedproducts 11 2.1. Equivariant Kasparov theory 11 2.2. Basicpropertiesofcrossedproducts 12 2.3. The dual action and Takai 13 2.4. Connes’ “Thom isomorphism” 13 2.5. The Pimsner-Voiculescu Theorem 16 2.6. The Baum-Connes Conjecture 16 2.7. The approach of Meyer and Nest 18 Lecture 3. The universal coefficient theorem for KK and some of its applications 19 3.1. Introduction to the UCT 19 3.2. TheproofofRosenbergandSchochet 19 3.3. The equivariant case 20 3.4. The categorical approach 20 3.5. The -theoretic approach 22 3.6. Topological applications 22 3.7. Applications to C∗-algebras 23

Date: 2010. 2010 Subject Classification. Primary 46L85; Secondary 19-02 46L87 19K35. Preparation of these lectures was partially supported by NSF grant DMS-0805003 and DMS- 1041576. The author thanks the members of the GNC Group at Buenos Aires, and especially Guillermo Corti˜nas, for their excellent organization of the Winter School. 1 2 JONATHAN ROSENBERG

Lecture 4. A fundamental example in noncommutative geometry: and geometry of the irrational rotation algebra 24 4.1. Basic facts about Aθ 24 ∞ 4.2. Basic facts about Aθ 26 4.3. Geometry of vector bundles 27 4.4. Miscellaneous other facts about Aθ 28 Lecture 5. Applications of the irrational rotation algebra in and physics 29 5.1. Applications to number theory 29 5.2. Applications to physics 29 References 30 APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 3

Lecture 1. Introduction to Kasparov’s KK-theory 1.1. Why KK? KK-theory is a bivariant version of topological K-theory, defined for C∗-algebras, with or without a . It can be defined for either real or complex algebras, but in these notes we will stick to complex algebras for sim- plicity. Thus if A and B are complex C∗-algebras, subject to a minor technical requirement (that B be σ-unital, which is certainly the case if it is either uni- tal or separable), an abelian group KK(A, B) is defined, with the property that KK(C,B)= K(B)= K0(B) if the first algebra A is just the scalars. (For the basic properties of K0, I refer you to the courses by Reich and Karoubi.) The theory was defined by Gennadi Kasparov in a remarkable series of papers: [33, 34, 35]. However, the definition at first seems highly technical and unmotivated, so it’s worth first seeing where the theory comes from and why one might be interested in it. For purposes of this introduction, we will only be concerned with the case where A and B are commutative. Thus A = C0(X) and B = C0(Y ), where X and Y are locally compact Hausdorff spaces. We will abbreviate KK(C0(X), C0(Y )) to KK(X, Y ). It is worth pointing out that the study of KK(X, Y ) (without considering KK(A, B) more generally) is already highly nontrivial, and encom- passes most of the features of the general theory. Note that we expect to have KK(C, C0(Y )) = KK(pt, Y ) = K(Y ), the K-theory of Y with compact support. Recall that this is the Grothendieck group of complexes of vector bundles that are exact off a compact set. It’s actually enough to take complexes of length 2, so an element of K(Y ) is represented by a pair of vector bundles V and V ′ over Y , ϕ together with a morphism of vector bundles V −→ V ′ that is an isomorphism off a compact set. Alternatively, K(Y ) can be identified with the reduced K-theory K(Y+) of the one-point compactification Y+ of Y . A good place to start in trying to understand KK is Atiyah’s paper [4] on thee Bott periodicity theorem. Bott periodicity, or more generally, the Thom iso- morphism theorem for a complex , asserts that if p: E → X is a complex vector bundle (more generally, one could take an even-dimensional real vector bundle with a spinc structure), then there is a natural isomorphism βE : K(X) → K(E), called the Thom isomorphism in K-theory. In the special case where X = pt, E is just Cn for some n, and we are asserting that there is a natural isomorphism Z = K(pt) → K(Cn) = K(R2n) = K(S2n), the Bott pe- ∗ riodicity map. The map βE can be described by the formula βE(a) = p (a) · τE. Here p∗(a) is the pull-back of a ∈ K(X) to E. Since a had compacte support, p∗(a) has compact support in the base direction of E, but is constant on fibers of p, so it certainly does not have compact support in the fiber direction. However, we can multiply it by the Thom class τE, which does have compact support along the fibers, and the product will have compact support in both directions, and will thus give a class in K(E) (remember that since E is necessarily noncompact, assuming n > 0, we need to use K-theory with compact support). The Thom class τE, in turn, can be described [59, §3] as an explicit complex • p∗E over E. The vector bundles in this complex are the exterior powers of E pulled back from X to E, and j j+1 V the map at a point e ∈ Ex from Ex to Ex is simply exterior product with e. This complex has compact support in the fiber directions since it is exact off the zero-section of E. (If e 6= 0, thenV the kernelV of e ∧ is spanned by products e ∧ ω.) 4 JONATHAN ROSENBERG

So far this is all simple vector bundle theory and KK is not needed. But it comes in at the next step. How do we prove that βE is an isomorphism? The simplest way would be to construct an inverse map αE : K(E) → K(X). But there is no easy way to describe such a map using topology alone. As Atiyah recognized, the easiest way to construct αE uses elliptic operators, in fact the family of Dolbeault operators along the fibers of E. Thus whether we like it or not, some analysis comes in at this stage. In more modern language, what we really want is the class αE in KK(E,X) corresponding to this family of operators, and the verification of the Thom isomorphism theorem is a Kasparov product calculation, the fact that αE is a KK inverse to the class βE ∈ KK(X, E) described (in slightly other terms) before. Atiyah also noticed [4] that it’s really just enough (because of certain identities about products) to prove that αE is a one-way inverse to βE, or in other words, in the language of Kasparov theory, that βE ⊗E αE = 1X . This comes down to an index calculation, which because of naturality comes down to the single calculation β ⊗C α = 1 ∈ KK(pt, pt) when X is a point and E = C, which amounts to the Riemann-Roch theorem for CP1. What then is KK(X, Y ) when X and Y are locally compact spaces? An element of KK(X, Y ) defines a map of K-groups K(X) → K(Y ), but is more than this; it is in effect a natural family of maps of K-groups K(X × Z) → K(Y × Z) for arbitrary Z. Naturality of course means that one gets a natural transformation of functors, from Z 7→ K(X × Z) to Z 7→ K(Y × Z). (Nigel Higson has pointed out that one can use this in reverse to define KK(X, Y ) as a natural family of maps of K-groups K(X × Z) → K(Y × Z) for arbitrary Z. The reason why this works will be explained in Lecture 3 in this series.) In particular, since KK(X × Rj ) can be identified with K−j(X), an element of KK(X, Y ) defines a graded map of K-groups Kj(X) → Kj(Y ), at least for j ≤ 0 (but then for arbitrary j because of Bott periodicity). The example of Atiyah’s class αE ∈ KK(E,X), based on a family of elliptic operators over E parametrized by X, shows that one gets an element of the bivariant K-group KK(X, Y ) from a family of elliptic operators over X parametrized by Y . The element that one gets should be invariant under of such operators. Hence Kasparov’s definition of KK(A, B) is based on a notion of homotopy classes of generalized elliptic operators for the first algebra A, “parametrized” by the second algebra B (and thus commuting with a B-module structure).

1.2. Kasparov’s original definition. As indicated above in Section 1.1, an ele- ment of KK(A, B) is roughly speaking supposed to be a homotopy class of families of elliptic pseudodifferential operators over A parametrized by B. For technical reasons, it’s convenient to work with self-adjoint bounded operators1, but it’s well- known that the most interesting elliptic operators send sections of one vector bundle to sections of another. The way to get around this is to take our operators to be self-adjoint, but odd with respect to a grading, i.e., of the form 0 F ∗ (1.1) T = T ∗ = . F 0  

1Often we want to apply the theory to self-adjoint differential operators D, which are never 2 2 − 1 bounded on L spaces. The trick is to replace D by D(1 + D ) 2 , which has the same index theory as D and is bounded. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 5

The operator F here really does act between different spaces, but T , built from F and F ∗, is self-adjoint, making it easier to work with. Then we need various conditions on T that correspond to the terms “elliptic,” “pseudodifferential,” and “parametrized by B.” So this boils down to the following. A class in KK(A, B) is represented by a Kasparov A-B-bimodule, that is, a Z/2-graded (right) Hilbert B- module H = H0 ⊕H1, together with a B-linear operator T ∈ L(H) of the form (1.1), and a (grading-preserving) ∗-representation φ of A on H, subject to the conditions that φ(a)(T 2 − 1) ∈ K(H) and [φ(a),T ] ∈ K(H) for all a ∈ A. These conditions require a few comments. The condition that φ(a)(T 2 − 1) ∈ K(H) is “ellipticity” and the condition that [φ(a),T ] ∈ K(H) is “pseudolocality.” If B = C, a Hilbert B-module is just a Hilbert , L(H) is the set of bounded linear operators on H, and K is the set of compact operators on H. If B = C0(Y ), a Hilbert B- module is equivalent to a continuous field of Hilbert spaces over Y . In this case, K(H) is the associated set of continuous fields of compact operators, while L(H) consists of continuous fields (continuity taken in the strong-∗ operator topology) of bounded operators. If X is another locally , then it is easy to see that Kasparov’s conditions are an abstraction of a continuous family of elliptic pseudolocal Hilbert space operators over X, parametrized by Y . Finally, if B is arbitrary, a Hilbert B-module means a right B-module equipped with a B-valued inner product h , iB, right B-linear in the second variable, satisfying ∗ hξ, ηiB = hη, ξiB and hξ, ξiB ≥ 0 (in the sense of self-adjoint elements of B), with equality only if ξ = 0. Such an inner product gives rise to a on H: 1/2 kξk = khξ, ξiB kB , and we require H to be complete with respect to this norm. Given a Hilbert B-module H, there are two special C∗-algebras associated to it. The first, called L(H), consists of bounded B-linear operators a on H, admitting ∗ ∗ an adjoint a with the usual property that haξ,ηiB = hξ,a ηiB for all ξ, η ∈ H. Unlike the case where B = C, existence of an adjoint is not automatic, so it must be explicitly assumed. Then inside L(H) is the ideal of B-compact operators. This is the closed linear span of the “rank-one operators” Tξ,η defined by Tξ,η(ν)= ξhη, νiB . Note that ∗ hTξ,η(ν),ωiB = hξhη, νiB ,ωiB = hω, ξhη, νiB iB ∗ ∗ ∗ = (hω, ξiBhη, νiB ) = hη, νiB hω, ξiB

= hν, ηiBhξ,ωiB = hν, ηhξ,ωiB iB

= hν,Tη,ξ(ω)iB ,

∗ so that Tξ,η = Tη,ξ. It is also obvious that if a ∈ L(H), then aTξ,η = Taξ,η, while ∗ ∗ ∗ ∗ ∗ ∗ ∗ Tξ,ηa = Tη,ξ a = a Tη,ξ = Ta∗η,ξ = Tξ,a η, so these rank-one operators generate an ideal in L(H), which is just the usual ideal of compact operators in   case B = C. For more on Hilbert C∗-modules and the C∗-algebras acting on them, see [40] or [46, Ch. 2]. The simplest kind of Kasparov bimodule is associated to a homomorphism φ: A → B. In this case, we simply take H = H0 = B, viewed as a right B- ∗ module, with the B-valued inner product hb1,b2iB = b1b2, and take H1 = 0 and T = 0. In this case, L(H) = M(B) (the multiplier algebra of B, the largest C∗- algebra containing B as an essential ideal), and K(H) = B. So φ maps A into K(H), and even though T = 0, the condition that φ(a)(T 2 − 1) ∈ K(H) is satisfied for any a ∈ A. 6 JONATHAN ROSENBERG

One special case which is especially important is the case where A = B and φ is the identity map. The above construction then yields a distinguished element 1A ∈ KK(A, A), which will play an important role later. In applications to index theory, Kasparov A-B-bimodules typically arise from elliptic (or hypoelliptic) pseudodifferential operators. However, there are other ways to generate Kasparov bimodules which we will discuss in Section 1.4 below. So far we have explained what the cycles are for KK-theory, but not the equiv- alence relation that determines when two such cycles give the same KK-element. First of all, there is a natural associative addition on Kasparov bimodules, ob- tained by taking the direct sum of Hilbert B-modules and the block direct sum of homomorphisms and operators. Then we divide out by the equivalence relation generated by addition of degenerate Kasparov bimodules (those for which for all a ∈ A, φ(a)(T 2 − 1)=0 and[φ(a),T ] = 0) and by homotopy. (A homotopy of Kas- parov A-B-bimodules is just a Kasparov A-C([0, 1],B)-bimodule.) Then it turns out that the resulting semigroup KK(A, B) is actually an abelian group, with in- version given by reversing the grading, i.e., reversing the roles of H0 and H1, and interchanging F and F ∗. Actually, it was not really necessary to divide out by degenerate bimodules, since if (H,φ,T ) is degenerate, then (C0((0, 1], H) (along with the action of A and the operator which are given by φ and T at each point of (0, 1]) is a homotopy from (H,φ,T ) to the 0-module. An interesting exercise is to consider what happens when A = C and B is a unital ∗ C -algebra. Then if H0 and H1 are finitely generated projective (right) B-modules and we take T = 0 and φ to be the usual action of C by scalar multiplication, we get a Kasparov C-B-bimodule corresponding to the element [H0] − [H1] of K0(B). With some work one can show that this gives an isomorphism between the Grothendieck group K0(B) of usual K-theory and KK(C,B). By considering what happens when one adjoins a unit, one can then show that there is still a natural isomorphism between K0(B) and KK(C,B), even if B is nonunital. Another important special case is when A and B are Morita equivalent in the sense of Rieffel [47, 50] — see [46] for a very good textbook treatment. That means we have an A-B-bimodule X with the following special properties:

(1) X is a right Hilbert B-module and a left Hilbert A-module. (2) The left action of A is by bounded adjointable operators for the B-valued inner product, and the right action of B is by bounded adjointable operators for the A-valued inner product. (3) The A- and B-valued inner products on X are compatible in the sense that if ξ,η,ν ∈ X, then Ahξ, ηiν = ξhη, νiB . (4) The inner products are “full,” in the sense that the image of Ah , i is dense in A, and the image of h , iB is dense in B.

Under these circumstances, X defines classes in [X] ∈ KK(A, B) and [X] ∈ KK(B, A) which are inverses to each other (with respect to the product discussed below in Section 1.3). Thus as far as KK-theory is concerned, A and B aree es- sentially equivalent. The construction of [X] and of [X] is fairly straightforward; for example, to construct [X], take H0 = X (viewed as a right Hilbert B-module), H1 = 0, and T = 0, and let φ: A → L(H) be the lefte action of A (which factors through L(H) by axiom (2)). By axiom (4) (which is really the key property), any element of A can be approximated by linear combinations of inner products Ahξ, ηi. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 7

For such an inner product, we have

φ(Ahξ, ηi)ν = ξhη, νiB = Tξ,η(ν), so the action of A on H is by operators in K(H), which is what is needed for the conditions for a Kasparov bimodule. The prototype example of a Morita equivalence has A = C, B = K(H) (we usually drop the H and just write K if the Hilbert space is infinite-dimensional and separable), and X = H, with the B-valued inner product taking a pair of vectors in H to the corresponding rank-one operator. Thus from the point of KK-theory, C and K are essentially indistinguishable, and so are B and B ⊗ K for any B. There is a converse [10]; separable C∗-algebras A and B are Morita equivalent if and only if A ⊗ K and B ⊗ K are isomorphic. (This condition, called stable isomorphism, is obviously satisfied by B and B ⊗ K, since (B ⊗ K) ⊗ K =∼ B ⊗ (K ⊗ K) =∼ B ⊗ K.) However, a Morita equivalence between A and B leads directly to a KK- equivalence, but not directly to an isomorphism A ⊗ K =∼ B ⊗ K (which requires some arbitrary choices). The most readable references for the material of this section are the book by Blackadar [6], Chapter VIII, and the “primer” of Higson [29].

1.3. Connections and the product. The hardest aspect of Kasparov’s approach to KK is to prove that there is a well-defined, functorial, bilinear, and associative product ⊗B : KK(A, B) × KK(B, C) → KK(A, C). There is also an external product ⊠: KK(A, B) × KK(C,D) → KK(A ⊗ C,B ⊗ D), where ⊗ denotes the completed tensor product. (For our purposes, the minimal or spatial C∗-tensor product will suffice.) The external product is actually built from the usual product using an operation called dilation (external product with 1). We can dilate a class a ∈ KK(A, B) to a class a ⊠ 1C ∈ KK(A ⊗ C,B ⊗ C), by taking a representative (H,φ,T ) for a to the bimodule (H⊗ C, φ ⊗ 1C,T ⊗ 1). Similarly, we can dilate a class b ∈ KK(C,D) (on the other side) to a class 1B ⊠ b ∈ KK(B ⊗ C,B ⊗ D). Then a ⊠ b = (a ⊠ 1C) ⊗B⊗C (1B ⊠ b) ∈ KK(A ⊗ C,B ⊗ D), and one can check that this is the same as what one gets by computing in the other order as (1A ⊠ b) ⊗A⊗D (a ⊠ 1D). The Kasparov products, as they are called, encompass the usual cup and cap products relating K-theory and K-. For example, the cup product in ordinary topological K-theory for a compact space X, ∪: K(X) × K(X) → K(X), is a composite of two products. Given a ∈ K(X)= KK(C, C(X)) and b ∈ K(X)= KK(C, C(X)), we first form the external product a ⊠ b ∈ KK(C, C(X) ⊗ C(X)) = KK(C, C(X × X)). Then we have

a ∪ b = (a ⊠ b) ⊗C(X×X) ∆, where ∆ ∈ KK(C(X × X), C(X)) is the class of the homomorphism defined by restriction of functions on X × X to the diagonal copy of X. In any event, it still remains to construct the product KK(A, B)×KK(B, C) → KK(A, C). Suppose we have classes represented by (E1, φ1,T1) and (E2, φ2,T2), where E1 is a right Hilbert B-module, E2 is a right Hilbert C-module, φ1 : A → L(E1), φ2 : B → L(E2), T1 essentially commutes with the image of φ1, and T2 essentially commutes with the image of φ2. It is clear that we want to construct the product using H = E1 ⊗B,φ2 E2 and φ = φ1 ⊗ 1: A → L(H). The main 8 JONATHAN ROSENBERG difficulty is getting the correct operator T . In fact there is no canonical choice; the choice is only unique up to homotopy. The most convenient method of doing the construction seems to be using the notion of a connection due to Connes and Skandalis [13], nicely explained in [6, §18] or [61]. To motivate this, let’s just consider a simple example that comes up in index theory, the construction of an “elliptic operator with coefficients in a vector bundle.” Let T be an elliptic operator on a compact manifold M, which we take to be a of the form (1.1) (acting on a Z/2-graded Hilbert space H), and let E be a complex vector bundle over M. Often we want to form TE, the same operator with coefficients in the vector bundle E. This is actually a special case of the Kasparov product. The sections Γ(M, E) are a finitely generated projective C(M)-module E; since C(M) is commutative, we can regard this as a C(M)-C(M)- bimodule, with the same action on the left and on the right. Then E (concentrated entirely in degree 0, together with the 0-operator), defines a KK-class [[E]] ∈ KK(M,M), while T defines a class [T ] in KK(M, pt). Note that forgetting the →֒ left C(M)-action on E is the same as composing with inclusion of the scalars C C(M) to get from [[E]] a class [E] ∈ KK(pt,M) = K(M), which is the usual K-theory class of E. The class of the operator TE will be the Kasparov product [[E]] ⊗M [T ] ∈ KK(M, pt). Defining the operator, however, requires a choice of connection on the bundle E. One way to get this is to embed E as a direct summand in a trivial bundle M × Cn. Then orthogonal projection onto E is given by a self- adjoint projection p ∈ C(M,Mn(C)). We can certainly form T ⊗ 1 acting on n H⊗ C , on which we have an obvious action of C(M) ⊗ Mn(C), but there is no reason why T ⊗ 1 and p should commute, so there is no “natural” cut-down of T to E. Thus we simply take the compression T ′ = p(T ⊗ 1)p acting on H′ = pH with the obvious action φ′ of C(M). Since T commutes with the action of C(M) up to compact operators, the commutator [p,T ⊗ 1] is also compact, so T ′ satisfies the requirements that (T ′)2 − 1 ∈ K(H′) and [φ′(f),T ′] ∈ K(H′). Its Kasparov class is well-defined, even though there is great freedom in choosing the operator (corresponding to the freedom to embed E in a trivial bundle in many different ways). (When n is large enough, all vector bundle embeddings of E into M × Cn are isotopic, and thus the operators obtained by the above construction will be homotopic in a way preserving the Kasparov requirements.) 1.4. Cuntz’s approach. Joachim Cuntz noticed in [17] that all Kasparov bimod- ules can be taken to come from the basic notion of a quasihomomorphism between C∗-algebras A and B. A quasihomomorphism A ⇉ D D B is roughly speaking a formal difference of two homomorphisms f± : A → D, neither of which maps into B itself, but which agree modulo an ideal isomorphic to B. Thus a 7→ f+(a) − f−(a) is a linear map A → B. Suppose for simplicity (one can always reduce to this case) that D/B =∼ A, so that f± are two splittings for an extension 0 → B → D → A → 0. Then for any split-exact functor F from C∗-algebras to abelian groups (meaning it sends split extensions to short exact sequences — an example would be F (A)= K(A ⊗ C) for some coefficient algebra C), we get an exact sequence

(f+)∗ / / r / / 0 / F (B) / F (D) l / F (A) / 0. (f−)∗ APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 9

Thus (f+)∗ − (f−)∗ gives a well-defined homomorphism F (A) → F (B), which we might well imagine should come from a class in KK(A, B). (Think about Section 1.1, where we mentioned Higson’s idea of defining KK(X, Y ) in terms of natural transformations of functors, from Z 7→ K(X × Z) to Z 7→ K(Y × Z). We will certainly get such a natural transformation from a quasihomomorphism C0(X) ⇉ D D C0(Y )⊗K, since C0(Y )⊗K and Y have the same K-theory.) And indeed, given a quasihomomorphism as above, we get a Kasparov A-B-bimodule, with B ⊕ B as the Hilbert B-module (with the obvious grading), with φ: A → L(B ⊕ B) defined by f 0 0 1 + , and T = . 0 f 1 0  −   The “almost commutation” relation is f (a) 0 0 1 0 f (a) − f (a) + , = + − ∈ K(B ⊕ B), 0 f (a) 1 0 f (a) − f (a) 0  −     − +  since K(B ⊕ B)= M2(B). In the other direction, given a Kasparov A-B-bimodule, one can add on a degenerate bimodule and do a homotopy to reduce it to something roughly of this form, showing that all of KK(A, B) comes from quasihomomor- phisms (see [6, §17.6]). The quasihomomorphism approach to KK makes it possible to define KK(A, B) in a seemingly simpler way [18]. To do this, Cuntz observed that a quasihomomor- phism A ⇉ D D B factors through a universal algebra qA constructed as follows. Start with the free product C∗-algebra QA = A∗A, the completion of linear combi- nations of words in two copies of A. There is an obvious surjective homomorphism QA ։ A obtained by identifying the two copies of A. The kernel of QA ։ A is called qA, and if

f+ / / u / / 0 / B / D i / A / 0 f− is a quasihomomorphism, we get a commutative diagram 0 / qA / QA / A / 0

  0 / B / D / A / 0, with the first copy of A in QA mapping to D via f+, and the second copy of A in QA mapping to D via f−. Thus homotopy classes of (strict) quasihomomorphisms from A to B can be identified with homotopy classes of ∗-homomorphisms from qA to B, and KK(A, B) turns out to be simply the set of homotopy classes of ∗-homomorphisms from qA to B ⊗ K. 1.5. Higson’s approach. There is still another very elegant approach to KK- theory due to Nigel Higson [28]. Namely, one can construct an additive category KK whose objects are the separable C∗-algebras, and where the morphisms from A to B are given by KK(A, B). Associativity and bilinearity of the Kasparov product, along with properties of the special elements 1A ∈ KK(A, A), ensure that this is indeed an additive category. What Higson did is to give an alternative construction of this category. Namely, start with the homotopy category of sep- arable C∗-algebras, where the morphisms from A to B are the homotopy classes 10 JONATHAN ROSENBERG of ∗-homomorphisms A → B. Then KK is the smallest additive category with the same objects, these morphisms, plus enough additional morphisms so that two basic properties are satisfied: (1) Matrix stability. If A is an object in KK (that is, a separable C∗-algebra) and if e is a rank-one projection in K = K(H), H a separable Hilbert space, then the homomorphism a 7→ a⊗e, viewed as an element of Hom(A, A⊗K), is an equivalence in KK, i.e., has an inverse in KK(A ⊗ K, A). s v (2) Split exactness. If 0 / A / B / C / 0 is a split short ex- act sequence of separable C∗-algebras, then for any separable C∗-algebra D, q s∗ 0 / KK(D, A) / KK(D,B) / KK(D, C) / 0 and ∗ q s 0 / KK(C,D) / KK(B,D) / KK(A, D) / 0 are split exact. Incidentally, if one just starts with the homotopy category and requires (1), matrix stability, that is already enough to guarantee that the resulting category has Hom-sets which are commutative monoids and that composition is bilinear [54, Theorem 3.1]. So it’s not asking much additional to require that one have an additive category. The proof of Higson’s theorem very much depends on the Cuntz construction in Section 1.4 above. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 11

Lecture 2. K-theory and KK-theory of crossed products 2.1. Equivariant Kasparov theory. Many of the interesting applications of KK- theory involve actions of groups in some way. For this, Kasparov also invented an equivariant version of the theory. In what follows, G will always be a second- countable locally compact group. A G-C∗-algebra will mean a C∗-algebra A, along with an action of G on A by ∗-automorphisms, continuous in the sense that the map G × A → A is jointly continuous. (Another way to say this is that if we give Aut A the topology of pointwise convergence, then G → Aut A is a continuous group homomorphism.) If G is compact, making the theory equivariant is rather straightforward. We just require all algebras and Hilbert modules to be equipped with G-actions, we require φ: A → L(H) to be G-equivariant, and we require the operator T ∈ L(H) to be G-invariant. This produces groups KKG(A, B) for (sep- arable, say) G-C∗-algebras A and B, and the same argument as before shows that G ∼ G KK (C,B) = K0 (B), equivariant K-theory. (In the commutative case, this is described in [59]. A general description may be found in [6, §11].) In particular, KKG(C, C) =∼ R(G), the representation ring of G, in other words, the Grothendieck group of the category of finite-dimensional representations of G, with product com- ing from the tensor product of representations. The rings R(G) are commutative, Noetherian if G is a compact Lie group, and often easily computable; for exam- ple, if G is compact and abelian, R(G) =∼ Z[G], the group ring of the Pontrjagin dual. If G is a compact connected Lie group with maximal torus T and Weyl group W W W = NG(T )/T , then R(G) =∼ R(T ) =∼ Z[T ]b . The properties of the Kasparov product all go through without change, since it is easy to “average” things with re- spect to a compact group action. Then Kasparovb product with KKG(C, C) makes all KKG-groups into modules over the ground ring R(G), so that homological alge- bra of the ring R(G) comes into play in understanding the equivariant KK-category KKG. When G is noncompact, the definition and properties of KKG are considerably more subtle, and were worked out in [35]. A shorter exposition may be found in [36]. The problem is that in this case, topological vector spaces with a continuous G-action are very rarely completely decomposable, and there are rarely enough G-equivariant operators to give anything useful. Kasparov’s solution was to work with G-continuous rather than G-equivariant Hilbert modules and operators; rather remarkably, these still give a useful theory with all the same formal properties as before. The KKG-groups are again modules over the R(G) = KKG(C, C), though this ring no longer has such a simple interpretation as before, and in fact, is not known for most connected semisimple Lie groups. A few functorial properties of the KKG-groups will be needed below, so we just mention a few of them. First of all, if H is a closed subgroup of G, then any G- C∗-algebra is by restriction also an H-C∗-algebra, and we have restriction maps KKG(A, B) → KKH (A, B). To go the other way, we can “induce” an H-C∗-alge- ∗ G bra A to get a G-C -algebra IndH (A), defined by

G IndH (A)= {f ∈ C(G, A) | f(gh)= h · f(g) ∀g ∈ G, h ∈ H, kf(g)k → 0 as g → ∞ mod H} .

G The induced action of G on IndH (A) is just left translation. An “imprimitivity G theorem” due to Green shows that IndH (A) ⋊ G and A ⋊ H are Morita equivalent. 12 JONATHAN ROSENBERG

If A and B are H-C∗-algebras, we then have an induction homomorphism H G G G KK (A, B) → KK (IndH (A), IndH (B)). The last basic operation on the KKG-groups depends on crossed products, so we consider these next.

2.2. Basic properties of crossed products. Suppose A is a G-C∗-algebra. Then one can define two new C∗-algebras, called the full and reduced crossed products of A by G, which capture the essence of the group action. These are easiest to define when G is discrete and A is unital. Then the full crossed product A ⋊α G (we often omit the α if there is no possibility of confusion) is the universal C∗-algebra generated by a copy of A and unitaries ug, g ∈ G, subject to the commutation ∗ condition ugaug = αg(a), where α denotes the action of G on A. The reduced crossed product A ⋊α,r G is the image of A ⋊α G in its “regular representation” π on L2(G, H), where H is a Hilbert space on which A acts faithfully, say by a representation ρ. Here A acts by (π(a)f)(g) = ρ(αg−1 (a))f(g) and G acts by left translation. The compatibility condition is satisfied since ∗ ′ ∗ −1 ′ π(ug)π(a)π(ug )f(g ) = (π(a)π(ug )f)(g g ) ∗ −1 ′ = ρ(αg′−1g(a))(π(ug )f)(g g ) ′ = ρ(αg′−1g(a))(f(g )) ′ ′ = ρ(αg′−1 (αg(a))(f(g )) = π(αg(a))f(g ). In the general case (where A is not necessarily unital and G is not necessarily discrete), the full crossed product is still defined as the universal C∗-algebra for covariant pairs of a ∗-representation ρ of A and a unitary representation π of G, −1 satisfying the compatibility condition π(g)ρ(a)π(g ) = ρ(αg(a)). It may be con- structed by defining a convolution multiplication on Cc(G, A) and then completing ∗ in the greatest C -algebra norm. The reduced crossed product A ⋊α,r G is again 2 the image of A ⋊α G in its “regular representation” on L (G, H). For details of the construction, see [44, §7.6] and [62, Ch. 2]. If A = C, the crossed product A ⋊ G is simply the universal C∗-algebra for ∗ ∗ ∗ unitary representations of G, or the group C -algebra C (G), and A⋊r G is Cr (G), the image of C∗(G) in the left regular representation on L2(G). The natural map ∗ ∗ C (G) ։ Cr (G) is an isomorphism if and only if G is amenable. When the action α is trivial (factors through the trivial group {1}), then A⋊G is the maximal tensor ∗ ∗ product A ⊗max C (G) while A ⋊r G is the minimal tensor product A ⊗ Cr (G). ∗ ∗ Again, the natural map from A ⊗max C (G) to A ⊗ Cr (G) is an isomorphism if and only if G is amenable. When A and the action α are arbitrary, the natural map A⋊α G ։ A⋊α,r G is an isomorphism if G is amenable, but also more generally if the action α is amenable in a certain sense. For example, if X is a locally compact G-space, the action is automatically amenable if it is proper, whether or not G is amenable. A good short survey of amenability for group actions may be found in [1]. When X is a locally compact G-space, the crossed product C0(G) ⋊ G often serves as a good substitute for the “quotient space” X/G in cases where the latter is badly behaved. Indeed, if G acts freely and properly on X, then C0(X) ⋊ G is Morita equivalent to C0(X/G). If G acts locally freely and properly on X, then C0(X) ⋊ G is Morita equivalent to an “orbifold algebra” that encompasses APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 13 not only the topology of X/G but also the finite isotropy groups. But if the G- action is not proper, X/G may be highly non-Hausdorff, while C0(X) ⋊ G may be a perfectly well-behaved noncommutative algebra. A key case later on will the one where X = T is the circle group, G = Z, and the generator of G acts by multiplication by e2πiθ. When θ is irrational, every orbit is dense, so X/G is an indiscrete space, and C(T) ⋊ Z is what’s usually denoted Aθ, an irrational rotation algebra or noncommutative 2-torus. Now we can explain the relationships between equivariant KK-theory and crossed products. One connection is that if G is discrete and A is a G-C∗-algebra, there is a natural isomorphism KKG(A, C) =∼ KK(A ⋊ G, C). Dually, if G is compact, there is a natural Green-Julg isomorphism [6, §11.7] KKG(C, A) =∼ KK(C, A ⋊ G). Still another connection is that there is (for arbitrary G) a functorial homomorphism j : KKG(A, B) → KK(A ⋊ G, B ⋊ G) sending (when B = A) 1A to 1A⋊G. (In fact, j can be viewed as a functor from the equivariant Kasparov category KKG to the non-equivariant Kasparov category KK. Later we will study how close it is to being faithful.) There is also a variant of j using reduced crossed products, denoted jr [35, §3.11]. If B = C and G is discrete, then j can be identified with the map KK(A⋊G, C) → KK(A⋊G, C∗(G)) induced by the inclusion of scalars C ֒→ C∗(G). (The fact that G is discrete means that C∗(G) is unital.) The map j is split injective in this case since it is split by the map induced by C∗(G) → C, corresponding to the trivial representation of G. Similarly, if G is compact, then via Green-Julg, j can be identified with the map KK(C, A ⋊ G) → KK(C∗(G), A ⋊ G) induced by the map C∗(G) → C corresponding to the trivial representation of G. This is again a split injection since C∗(G) splits as the direct sum of C and summands associated to other representations.

2.3. The dual action and Takai duality. When the group G is not just locally compact but also abelian, then it has a Pontrjagin dual group G. In this case, given any G-C∗-algebra algebra A, say with α denoting the action of G on A, there is a dual action α of G on the crossed product A ⋊ G. When Ab is unital and G is discrete, so that A ⋊ G is generated by a copy of A and unitaries ug, g ∈ G, the dual action is givenb b simply by

αγ (aug)= aughg,γi.

The same formula still applies in general, except that the elements a and ug don’t quite live in the crossed productb but in a larger algebra. The key fact about the 2 dual action is the Takai duality theorem: (A ⋊α G) ⋊αb G =∼ A ⊗ K(L (G)), and the ˜ double dual action αˆ of G˜ =∼ G on this algebra can be identified with α ⊗ Ad λ, where λ is the left regular representation of G on L2(G).b Good expositions may be found in [44, §7.9] and in [62, Ch. 7].

2.4. Connes’ “Thom isomorphism”. Recall that the Thom isomorphism the- orem in K-theory (see Section 1.1) asserts that if E is a complex vector bundle over X, there is an isomorphism of K-groups K(X) → K(E), implemented by a KK-class in KK(X, E). Now if Cn (or R2n — there is no difference since we are just considering the additive group structure) acts on X by a trivial ac- n ∗ n n tion α, then C0(X) ⋊α C =∼ C0(X) ⊗ C (C ) =∼ C0(X) ⊗ C0(C ) =∼ C0(E),

b 14 JONATHAN ROSENBERG where E is a trivial rank-n complex vector bundle over X. (We have used Pon- trjagin duality and the fact that abelian groups are amenable.) It follows that n n K(C0(X)) =∼ K(C0(X) ⋊α C ). Since any action α of C is homotopic to the trivial action and “K-theory is supposed to be homotopy invariant,” that suggests n ∗ that perhaps KK(A) =∼ KK(A ⋊α C ) for any C -algebra A and for any action α of Cn. This is indeed true and the isomorphism is implemented by classes (which n n are inverse to one another) in KK(A, A ⋊α C ) and KK(A ⋊α C , A). It is clearly enough to prove this in the case n = 1, since we can always break a crossed product by Cn up as an n-fold iterated crossed product. That A and A ⋊α C are always KK-equivalent or that they at least have the same K-theory, or (this is equivalent since one can always suspend on both sides) that A ⊗ C0(R) and A ⋊α R are always KK-equivalent or that they at least have the same K-theory for any action of R, is called Connes’ “Thom isomorphism” (with the name “Thom” in quotes since the only connection with the classical Thom isomorphism is the one we have already explained). Connes’ original proof is relatively elementary, but only gives an isomorphism of K-groups, not a KK- equivalence, and can be found in [12] or in [20, §10.2]. To illustrate Connes’ idea, let’s suppose A is unital and we have a class in K0(A) represented by a projection p ∈ A. (One can always reduce to this special case.) If α were to fix p, then 1 7→ p gives an equivariant map from C to A and thus would induce a map of crossed products C⋊R =∼ C0(R) → A ⋊α R or C⋊C =∼ C0(C) → A ⋊α C giving a map on K-theory β : Z → K0(A ⋊C). The image of [p] under the isomorphism K0(A) → K0(A⋊C) will be β(1).b So the idea is to show that oneb can modify the action to one fixing p (using a cocycle conjugacy) without changing the isomorphism class of the crossed product. There are now quite a number of proofs of Connes’ theorem available, each using somewhat different techniques. We just mention a few of them. A proof using K- theory of Wiener-Hopf extensions is given in [49]. There are also fancier proofs using KK-theory. If α is a given action of R on A and if β is the trivial action, one can try to construct KKR elements c ∈ KKR((A, α), (A, β)) and d ∈ KKR((A, β), (A, α)) R which are inverses of each other in KK . Then the morphism j of Section 2.1 sends these to KK-equivalences j(c) and j(d) between A ⋊α R and A ⋊β R =∼ A ⊗ C0(R). Another rather elegant approach, using KK-theory but not the equivariant groups, may be found in [26]. Fack and Skandalis use the group KK1(A, B), which we have avoided so far in order to simplify the theory, but it can be defined with triples (H,φ,T ) like those used for KK(A, B), but with two modifications:

(1) H is no longer graded, and there is no grading condition on φ. (2) T is self-adjoint but with no grading condition, and φ(a)(T 2 − 1) ∈ K(H) and [φ(a),T ] ∈ K(H) for all a ∈ A.

1 It turns out that KK (A, B) =∼ KK(A⊗C0(R),B), and that the Kasparov product can be extended to a graded commutative product on the direct sum of KK = KK0 and KK1. The product of two classes in KK1 can by Bott periodicity be taken to land in KK0. We can now explain the proof of Fack and Skandalis as follows. They show that for each separable C∗-algebra A with an action α of R, there is a special element 1 tα ∈ KK (A, A⋊α R) (constructed using a singular integral operator). Note by the way that doing the construction with the dual action and applying Takai duality APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 15

1 gives tαb ∈ KK (A⋊α R, A), since (A⋊α R)⋊αbR =∼ A⊗K, which is Morita equivalent to A. These elements have the following properties:

(1) (Normalization) If A = C (so that necessarily α = 1 is trivial), then t1 ∈ 1 KK (C, C0(R)) is the usual generator of this group (which is isomorphic to Z). (2) (Naturality) The elements are natural with respect to equivariant homo- morphisms ρ: (A, α) → (C,γ), in that ifρ ¯ denotes the induced map on ∗ crossed products, thenρ ¯∗(tα) = ρ (tγ ) ∈ KK(A, C ⋊γ R), and similarly, ∗ ρ¯ (tγb)= ρ∗(tαb) ∈ KK(A ⋊α R, C). (3) (Compatibility with external products) Given x ∈ KK(A, B) and y ∈ KK(C,D),

(tb ⊗ x) ⊠ y = t \ ⊗ (x ⊠ y). α A α⊗1C A⊗C Similarly, given x ∈ KK(B, A) and y ∈ KK(D, C),

y ⊠ (x ⊗A tα) = (y ⊠ x) ⊗C⊗A t1C ⊗α. 

Theorem 2.1 (Fack-Skandalis [26]). These properties completely determine tα, and tα is a KK-equivalence (of degree 1) between A and A ⋊α R.

Proof. Suppose we have elements tα satisfying the properties above. Let us first s s show that tα⊗A⋊αRtαb =1A. For s ∈ R, let α be the rescaled action αt = αst. Then s define an action β of R on B = C([0, 1], A) by (βtf)(s)= αt (f(s)). Let gs : B → A be evaluation at s, which is clearly an equivariant map (B,β) ։ (A, αs). We also get maps gs : B ⋊β R → A⋊αs R, and the double dual map gˆs can be identified with ∗ ∗ s b bs gs ⊗1: B ⊗K → A⊗K. By Axiom (2), (¯gs)∗(tβ)= gs (tα ) and (gs)∗(tβ)=¯gs (tα ). s s Let σs = tbα ⊗A⋊αs R tαb ∈ KK(A, A). By associativity of Kasparov products,

⋊ R b ⋊ R b (gs)∗ tβ ⊗B β tβ = tβ ⊗B β tβ ⊗B [gs]

s  = tβ ⊗B⋊βR [¯gs] ⊗A⋊αsR tαb

⋊ R ⋊ R bs = tβ ⊗B β [¯gs] ⊗A αs tα

s ⋊ R bs = [gs] ⊗A tα ⊗A αs tα = [gs] ⊗A σs. 

Since gs is a homotopy of maps B → A and KK is homotopy-invariant, [gs] = [g0]. But g0 is a homotopy equivalence with homotopy inverse f : a 7→ a ⊗ 1, so we see that ⋊ R b σs = [f] ⊗B tβ ⊗B β tβ ⊗B [g0] is independent of s. In particular, σ1 = tα ⊗A⋊αRtαb agrees with σ0, which can be computed to be 1A by Axioms (1) and (3) since the action of R is trivial in this case. So tα ⊗A⋊αR tαb = 1A. Replacing α by α and using Takai duality, this also implies that tαb ⊗A tα =1A⋊αR. So tα and tαb give KK-equivalences. The uniqueness falls out at the same time,b since we see from the above that s s [gs] ⊗A tα = tβ ⊗B⋊β R [¯gs] ∈ KK(B, A ⋊α R), and that all the KK-elements involved are KK-equivalences. Furthermore, we know by Axioms (1) and (3) that 1 tα0 = 1A ⊠ t1, where t1 is the special element of KK (C, C0(R)) mentioned in 0 Axiom (1). This determines tβ (from the identity [g0] ⊗A tα = tβ ⊗B⋊βR [¯g0]), and then tα is determined from the identity [g0] ⊗A tα = tβ ⊗B⋊βR [¯g1].  16 JONATHAN ROSENBERG

2.5. The Pimsner-Voiculescu Theorem. Connes’ Theorem from Section 2.4 computes K-theory or KK-theory for crossed products by R. This can be used to compute K-theory or KK-theory for crossed products by Z, using the fact from Section 2.2 that if A is a C∗-algebra equipped with an action α of Z (or equivalently, a single ∗-automorphism θ, the image of 1 ∈ Z under the action), then A ⋊α Z is R R Morita equivalent to IndZ(A, α) ⋊R. The algebra Tθ = IndZ(A, α) is often called the mapping torus of (A, θ); it can be identified with the algebra of continuous functions f : [0, 1] → A with f(1) = θ(f(0)). It comes with an obvious short exact sequence

0 → C0((0, 1), A) → Tθ → A → 0, for which the associated exact sequence in K-theory has the form

1−θ∗ 1−θ∗ ···→ K1(A) −−−→ K1(A) → K0(Tθ) → K0(A) −−−→ K0(A) →··· . Since

K0(A ⋊α Z) =∼ K0(Tθ ⋊Ind α R) =∼ K1(Tθ), and similarly for K0, we obtain the Pimsner-Voiculescu exact sequence

1−θ∗ ···→ K1(A) −−−→ K1(A) → K1(A ⋊α Z) → (2.1) 1−θ∗ → K0(A) −−−→ K0(A) → K0(A ⋊α Z) →··· .

Here one can check that the maps Kj(A) → Kj (A⋊αZ) are induced by the inclusion of A into the crossed product. For another proof, closer to the original argument of Pimsner and Voiculescu, see [20, Ch. 5].

2.6. The Baum-Connes Conjecture. The theorems of Connes and Pimsner- Voiculescu on K-theory of crossed products by R and Z suggest the question of whether there are similar results for other groups G. In particular, one would like ∗ to know if the K-theory of Cr (G), or better still, the K-theory of reduced crossed products A ⋊ G, can be computed in a “topological” way. The answer in many cases seems to be “yes,” and the conjectured answer is what is usually called the Baum-Connes Conjecture, with or without coefficients. The special case of the Baum-Connes Conjecture (without coefficients) for connected Lie groups is also known as the Connes-Kasparov Conjecture, and is now a known theorem. The Baum-Connes conjecture also has other origins, such as the Novikov Con- jecture on higher signatures and conjectures about algebraic K-theory of group rings, which will be touched on in Reich’s lectures. These other motivations for the conjecture mostly concern the case where G is discrete, which is actually the most interesting case of the conjecture, though there are good reasons for not restricting only to this case. (For example, as we already saw in the case of Z, information about discrete groups can often be obtained by embedding them in a Lie group.) Here is the formal statement of the conjecture. Conjecture 2.2 (Baum-Connes). Let G be a locally compact group, second-coun- table for convenience. Let EG be the universal proper G-space. (This is a con- tractible space on which G acts properly, characterized [5] up to G-homotopy equiv- alence by two properties: that every compact subgroup of G has a fixed point in EG, and that the two projections EG × EG → EG are G-homotopic. Here the product APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 17 space is given the diagonal G-action. If G has no compact subgroups, then EG is the usual universal free G-space EG.) There is an assembly map lim KG(X) → K (C∗(G)) −→ ∗ ∗ r X⊆EG X/G compact defined by taking G-indices of G-invariant elliptic operators, and this map is an isomorphism. Conjecture 2.3 (Baum-Connes with coefficients). With notation as in Conjecture 2.2, if A is any separable G-C∗-algebra, the assembly map lim KKG(C (X), A) → K (A ⋊ G) −→ ∗ 0 ∗ r X⊆EG X/G compact is an isomorphism. Let’s see what the conjecture amounts to in some special cases. If G is compact, EG can be taken to be a single point. The conjecture then asserts that the assembly G ∗ ∗ map KK∗ (pt, pt) → K∗(C (G)) is an isomorphism. For G compact, C (G) is by the Peter-Weyl Theorem the completed direct sum of matrix algebras V End(V ), where V runs over a set of representatives for the irreducible representations of ∗ L ∗ G. Thus K1(C (G)) (remember this is topological K1) vanishes and K0(C (G)) =∼ R(G). The assembly map in this case is the Green-Julg isomorphism of Section G 2.2. In fact, the same holds with coefficients; the assembly map KK∗ (C, A) = G K∗ (A) → K∗(A ⋊ G) is the Green-Julg isomorphism, and Conjecture 2.3 is true. Next, suppose G = R. Since G has no compact subgroups and is contractible, we can take EG = R with R acting on itself by translations. If A is an R-C∗-algebra, R the assembly map is a map KK∗ (C0(R), A) → K∗(A ⋊R). This map turns out to be Kasparov’s morphism R ∼ j : KK∗ (C0(R), A) → KK∗(C0(R) ⋊R, A ⋊R)= KK∗(K, A ⋊R) = K∗(A ⋊R), which is the isomorphism of Connes’ Theorem (Section 2.4). (The isomorphism C0(R) ⋊R =∼ K is a special case of the Imprimitivity Theorem giving a Morita G equivalence between Ind{1} A ⋊ G and A, or if you prefer, of Takai duality from Section 2.3.) So again the conjecture is true. Another good test case is G = Z. Then EG = EG = R, with Z acting by translations and quotient space T. The left-hand side of the conjecture is thus Z KK (C0(R), A), while the right-hand side is K(A ⋊ Z), which is computed by the Pimsner-Voiculescu sequence. More generally, suppose G is discrete and torsion-free. Then EG = EG, and the quotient space EG/G is the usual BG. The assembly map cmpct ∗ (for the conjecture without coefficients) maps K∗ (BG) → K∗(Cr (G)). (The left-hand side is K-homology with compact supports.) This map can be viewed as an index map, since classes in the K-homology group on the left are represented by generalized Dirac operators D over Spinc manifolds M with a G-covering, and the assembly map takes such an operator to its “Mishchenko-Fomenko index” with values in the K-theory of the (reduced) group C∗-algebra. The connection between this assembly map and the usual sort of assembly map studied by topologists is discussed in [55]. In particular, Conjecture 2.2 implies a strong form of the Novikov Conjecture for G. 18 JONATHAN ROSENBERG

2.7. The approach of Meyer and Nest. An interesting alternative approach to the Baum-Connes Conjecture has been proposed by Meyer and Nest [42, 43]. This approach is also briefly sketched (in somewhat simplified form) in [20, §5.3] and in [56, Ch. 5]. Meyer and Nest begin by observing that the equivariant KK- category, KKG, naturally has the structure of a triangulated category. It has a distinguished class E of weak equivalences, morphisms f ∈ KKG(A, B) which re- strict to equivalences in KKH(A, B) for every compact subgroup H of G. (Note that if G has no nontrivial compact subgroups, for example if G is discrete and torsion-free, then this condition just says that f is a KK-equivalence after forget- ting the G-equivariant structure.) The Baum-Connes Conjecture with coefficients, Conjecture 2.3, basically amounts to the assertion that if f ∈ KKG(A, B) is in E, 2 then jr(f) ∈ KK(A ⋊r G, B ⋊r G) is a KK-equivalence. In particular, suppose G has no nontrivial compact subgroups and satisfies Conjecture 2.3. Then if A is a G-C∗-algebra which, forgetting the G-action, is contractible, then the unique G morphism in KK (0, A) is a weak equivalence, and so (applying jr), the unique morphism in KK(0, A ⋊r G) is a KK-equivalence. Thus A ⋊r G is K-contractible, i.e., all of its topological K-groups must vanish. When G = R, this follows from Connes’ Theorem, and when G = Z, this follows from the Pimsner-Voiculescu exact sequence, (2.1). Now that we have several different formulations of the Baum-Connes Conjecture, it is natural to ask how widely the conjecture is valid. Here are some of the things that are known: (1) There is no known counterexample to Conjecture 2.2 (Baum-Connes for groups, without coefficients). Counterexamples are now known [27] to Con- jecture 2.3 with G discrete and A even commutative. (2) Conjecture 2.3 is true if G is amenable, or more generally, if it is a-T- menable, that is, if it has an affine, isometric and metrically proper action on a Hilbert space [30]. Such groups include all Lie groups whose noncompact semisimple factors are all locally isomorphic to SO(n, 1) or SU(n, 1) for some n. (3) Conjecture 2.2 is true for connected reductive Lie groups, connected reduc- tive p-adic groups, for hyperbolic discrete groups, and for cocompact lattice subgroups of Sp(n, 1) or SL(3, C) [39]. There is now a vast literature on this subject, but our intention here is not to be exhaustive, but just to give the reader some flavor of what’s going on.

2The reason for using jr in place of j for can be seen from the case of G nonamenable with property T. In this case, C∗(G) has a projection corresponding to the trivial representation of G ∗ which is “isolated,” and thus maps to 0 in Cr (G). So these two algebras do not have the same ∗ K-theory. It turns out at least in many examples that K0(Cr (G)) can be described in purely ∗ topological terms, but K0(C (G)) cannot. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 19

Lecture 3. The universal coefficient theorem for KK and some of its applications 3.1. Introduction to the UCT. Now that we have discussed KK and KKG, a natural question arises: how computable are they? In particular, is KK(A, B) G G determined by K∗(A) and by K∗(B)? Is KK (A, B) determined by K∗ (A) and G by K∗ (B)? A first step was taken by Kasparov [34]: he pointed out that KK(X, Y ) is given by an explicit topological formula when the one-point compactifications X+ and Y+ are finite CW complexes: KK(X, Y ) =∼ K(Y+ ∧ D(X+)), where D(X+) denotes 3 the Spanier-Whitehead dual of X+. Let’s make a definition — we say the paire of C∗-algebras (A, B) satisfies the Universal Coefficient Theorem for KK (or UCT for short) if there is an exact sequence

1 (3.1) 0 → ExtZ(K∗(A),K∗+1(B)) → KK(A, B) Z ∗∈M/2 ϕ −→ HomZ(K∗(A),K∗(B)) → 0. Z ∗∈M/2 Here ϕ sends a KK-class to the induced map on K-groups. We need one more definition. Let B be the bootstrap category, the smallest full subcategory of the separable C∗-algebras (with the ∗-homomorphisms as mor- phisms) containing all separable type I algebras, and closed under extensions, countable C∗-inductive limits, and KK-equivalences. Note that KK-equivalences include Morita equivalences, and type I algebras include commutative algebras. Recall from Section 1.2 that stably isomorphic separable C∗-algebras are Morita equivalent, hence KK-equivalent. Furthermore, separable type I C∗-algebras are inductive limits of finite iterated extensions of stably commutative C∗-algebras [44, Ch. 6]. Thus we could just as well replace the words “type I” by “commutative” in the definition of B. Furthermore, any compact metric space is a countable limit of finite CW complexes. Dualizing, that means that any unital separable commutative C∗-algebra is a countable inductive limit (i.e., categorical colimit) of algebras of the form C(X), X a finite CW complex, and any separable commutative C∗-algebra is a countable inductive limit (i.e., colimit) of algebras of the form C0(X), X+ a finite CW complex. We will use this fact shortly. Theorem 3.1 (Rosenberg-Schochet [58]). The UCT holds for all pairs (A, B) with A an object in B and B separable. Unsolved problem: Is every separable nuclear C∗-algebra in B? Skandalis [60] showed that there are non-nuclear algebras not in B, for which the UCT fails.

3.2. The proof of Rosenberg and Schochet. First suppose K∗(B) is injective as a Z-module, i.e., divisible as an abelian group. Then HomZ( ,K∗(B)) is an ∗ exact functor, so A 7→ HomZ(K∗(A),K∗(B)) gives a theory on C - algebras. In particular, ϕ is a natural transformation of homology theories for spaces ∗ X 7→ KK∗(C0(X),B) X 7→ HomZ(K (X),K∗(B)) .

3Spanier-Whitehead duality basically interchanged homology and cohomology. ∧ denotes the , the product in the category of spaces with distinguished basepoint. 20 JONATHAN ROSENBERG

Since ϕ is an isomorphism for X = Rn by Bott periodicity, it is an isomorphism whenever X+ is a sphere, and thus (by the analogue of the Eilenberg-Steenrod uniqueness theorem for generalized homology theories) whenever X+ is a finite CW complex. We extend to arbitrary locally compact X by taking limits, and then to the rest of B, using the observations we made before the proof of the theorem. In order to know we can pass to countable inductive limits, we need one additional fact about KK, namely that it is “countably additive” (sends countable C∗-algebra direct sums in the first variable to products of abelian groups). This fact is not hard to check from Kasparov’s original definition. So the theorem holds when K∗(B) is injective. The rest of the proof uses an idea due to Atiyah [3], of geometric resolutions. The idea is that given arbitrary B, we can change it up to KK-equivalence so that it fits into a short exact sequence 0 → C → B → D → 0 for which the induced K-theory sequence is short exact:

K∗(B) ֌ K∗(D) ։ K∗−1(C) and K∗(D), K∗(C) are Z-injective. Then we use the theorem for KK∗(A, D) and KK∗(A, C), along with the long exact sequence in KK in the second variable, to get the UCT for (A, B). 

3.3. The equivariant case. If one asks about the UCT in the equivariant case, then the of the ground ring R(G) becomes relevant. This is not always well behaved, so as noticed by Hodgkin [31], one needs restrictions on G to get anywhere. But for G a connected compact Lie group with π1(G) torsion-free, R(G) has finite global dimension, and the spectral sequence one ends up with does converge to the right limit. Theorem 3.2 (Rosenberg-Schochet [57]). If G is a connected compact Lie group ∗ with π1(G) torsion-free, and if A, B are separable G-C -algebras with A in a suit- able bootstrap category containing all commutative G-C∗-algebras, then there is a convergent spectral sequence

p G G G ExtR(G)(K∗ (A),Kq+∗(A)) ⇒ KK∗ (A, B). The proof is more complicated than in the non-equivariant case, but in the same spirit. Also along the same lines, there is a UCT for KK of real C∗-algebras, due to Boersema [8]. The homological algebra involved in this case is appreciably more complicated than in the complex C∗-algebra case, and is based on ideas on Bousfield [9] on the classification of K-local spectra.

3.4. The categorical approach. The UCT implies a lot of interesting facts about the bootstrap category B. Here are a few examples. Theorem 3.3 (Rosenberg-Schochet [58]). Let A, B be C∗-algebras in B. Then A and B are KK-equivalent if and only if they have the isomorphic topological K-groups. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 21

Proof. ⇒ is trivial. So suppose K∗(A) =∼ K∗(B). Choose an isomorphism

ψ : K∗(A) → K∗(B). Since the map ϕ in the UCT (3.1) is surjective, ψ is realized by a class x ∈ KK(A, B) (not necessarily unique, but just pick one). Now consider the commutative diagram with exact rows ϕ / 1 / / / 0 / ExtZ(K∗+1(B),K∗(A)) / KK∗(B, A) / Hom(K∗(B),K∗(A)) / 0

∗ ∗ ∼= ψ x⊗B ∼= ψ    ϕ  / 1 / / / 0 / ExtZ(K∗+1(A),K∗(A)) / KK∗(A, A) / Hom(K∗(A),K∗(A)) / 0

By the 5-Lemma, Kasparov product with x is an isomorphism KK∗(B, A) → KK∗(A, A). In particular, there exists y ∈ KK(B, A) with x ⊗B y =1A. Similarly, there exists z ∈ KK(B, A) with z ⊗A x =1B. Then by associativity

z = z ⊗A (x ⊗B y) = (z ⊗A x) ⊗B y = y and we have a KK-inverse to x.  Corollary 3.4. We can also describe B as the smallest full subcategory of the separable C∗-algebras closed under KK-equivalence and containing the separable commutative C∗-algebras. A separable C∗-algebra A has the property that (A, B) satisfies the UCT for all separable C∗-algebras if and only if it lies in B. Proof. Let B′ be the smallest full subcategory of the separable C∗-algebras closed under KK-equivalence and containing the separable commutative C∗-algebras. By definition of B, B′ is a subcategory of B. But if A is in B, its K-groups are countable. For any countable groups G0 and G1, it is easy to construct a second-countable with these K-groups. So there is a separable commutative ∗ C -algebra C0(Y ) with K∗(C0(Y )) =∼ K∗(A) (just as abelian groups). By Theorem ′ 3.3, there is a KK-equivalence between A and C0(Y ), so A lies in B . As far as the last statement is concerned, one direction is the UCT itself. For the other direction, suppose that (A, B) satisfies the UCT for all separable C∗-algebras B. In particular, it holds for a commutative B with the same K-groups as A, and by the argument above, A is KK-equivalent to B, hence lies in B. 

Recall that KK(A, A) = EndKK(A) is a ring under Kasparov product. We can now compute the ring structure. Theorem 3.5 (Rosenberg-Schochet). Suppose A is in B. In the UCT sequence 1 ϕ 0 → ExtZ(Ki+1(A),Ki(A)) → KK(A, A) −→ End(Ki(A)) → 0, Z Z i∈M/2 i∈M/2 ϕ is a split surjective homomorphism of rings, and J = ker ϕ (the Ext term) is an ideal with J 2 =0.

Proof. Choose A0 and A1 commutative with K0(A0) =∼ K0(A), K1(A0) = 0, K0(A1)=0, K1(A1) =∼ K1(A). Then by Theorem 3.3, A0 ⊕A1 is KK-equivalent to A, and without loss of generality, we may assume we have an actual splitting A = A0 ⊕ A1. By the UCT, KK(A0, A0) =∼ End K0(A) and KK(A1, A1) =∼ End K1(A). So KK(A0, A0)⊕KK(A1, A1) is a subring of KK(A, A) mapping isomorphically under ϕ. This shows ϕ is split surjective. We also have J = KK(A0, A1) ⊕ 22 JONATHAN ROSENBERG

KK(A1, A0). If, say, x lies in the first summand and y in the second, then x ⊗A1 y induces the 0-map on K0(A) and so is 0 in KK(A0, A0) =∼ End(K0(A)). Similarly, ∼ y⊗A0 x induces the 0-map on K1(A) and so is 0 in KK(A1, A1) = End(K1(A)).  3.5. The homotopy-theoretic approach. There is a homotopy-theoretic ap- proach to the UCT that topologists might find attractive; it seems to have been discovered independently by several people (e.g., [11, 32] — see also the review of [11] in MathSciNet). Let A and B be C∗-algebras and let K(A) and K(B) be their topological K-theory spectra. These are module spectra over K = K(C), the usual of complex K-theory. Then we can define top KK (A, B)= π0(HomK K(A), K(B)) . This is again using ideas of Bousfield [9].  Theorem 3.6. There is a natural map KK(A, B) → KKtop(A, B), and it’s an isomorphism if and only if the UCT holds for the pair (A, B). Observe that KKtop(A, B) even makes sense for Banach algebras, and always comes with a UCT. We promised in the first lecture to show that defining KK(X, Y ) to be the set of natural transformations (Z 7→ K(X × Z)) (Z 7→ K(Y × Z)) indeed agrees with Kasparov’s KK(C0(X), C0(Y )). Indeed, Z 7→ K(X × Z) is ba- sically the cohomology theory defined by K(X), and Z 7→ K(Y ×Z) is similarly the cohomology theory defined by K(Y ). So the natural transformations (commuting top with Bott periodicity) are basically a model for KK (C0(X), C0(Y )).

3.6. Topological applications. The UCT can be used to prove facts about topo- logical K-theory which on their face have nothing to do with C∗-algebras or KK. For example, we have the following purely topological fact: Theorem 3.7. Let X and Y be locally compact spaces such that K∗(X) =∼ K∗(Y ) just as abelian groups. Then the associated K-theory spectra K(X) and K(Y ) are homotopy equivalent.

Proof. We have seen (Theorem 3.3) that the hypothesis implies C0(X) and C0(Y ) are KK-equivalent, which gives the desired conclusion. 

Note that this theorem is quite special to complex K-theory; it fails even for ordinary cohomology (since one needs to consider the action of the Steenrod alge- bra). Similarly, the UCT implies facts about cohomology operations in complex K- theory and K-theory mod p. For example, one has: Theorem 3.8 (Rosenberg-Schochet [58]). The Z/2-graded ring of homology opera- tions for K( ; Z/n) on the category of separable C∗-algebras is the exterior algebra over Z/n on a single generator, the Bockstein β. Theorem 3.9 (Araki-Toda [2], new proof by Rosenberg-Schochet in [58]). There are exactly n admissible multiplications on K-theory mod n. When n is odd, exactly one is commutative. When n =2, neither is commutative. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 23

3.7. Applications to C∗-algebras. Probably the most interesting applications of the UCT for KK are to the classification problem for nuclear C∗-algebras. The Elliott program (to quote M. Rørdam from his review of the Kirchberg-Phillips paper [37]) is to classify “all separable, nuclear C∗-algebras in terms of an invariant that has K-theory as an important ingredient.” Kirchberg and Phillips have shown how to do this for Kirchberg algebras, that is simple, purely infinite, separable and nuclear C∗-algebras. The UCT for KK is a key ingredient. Theorem 3.10 (Kirchberg-Phillips [37, 45]). Two stable Kirchberg algebras A and B are isomorphic if and only if they are KK-equivalent; and moreover every in- vertible element in KK(A, B) lifts to an isomorphism A → B. Similarly in the unital case if one keeps track of [1A] ∈ K0(A). We will not attempt to explain the proof of Kirchberg-Phillips, but it’s based on the idea that a KK-class is given by a quasihomomorphism, which under the specific hypotheses can be lifted to a true homomorphism. More recent results of a somewhat similar nature may be found in [22, 21, 41]. Given the Kirchberg-Phillips result, one is still left with the question of deter- mining when two Kirchberg algebras are KK-equivalent. But those of “Cuntz 4 type” (like On) lie in B, and Kirchberg and Phillips show that ∀ abelian groups G0 and G1 and ∀g ∈ G0, there is a nonunital Kirchberg algebra A ∈B with these K-groups, and there is a unital Kirchberg algebra A ∈B with these K-groups and with [1A]= g. By the UCT, these algebras are classified by their K-groups. The original work on the Elliott program dealt with the opposite extreme: stably finite algebras. Here again, KK can play a useful role. Here is a typical result from the vast literature: Theorem 3.11 (Elliott [23]). If A and B are C∗-algebras of real rank 0 which are inductive limits of certain “basic building blocks”, then any x ∈ KK(A, B) preserving the “graded dimension range” can be lifted to a ∗-homomorphism. If x is a KK-equivalence, it can be lifted to an isomorphism.

This theorem applies for example to the irrational rotation algebras Aθ, because of an amazing result by Elliott and Evans [24] that shows that these algebras are indeed inductive limits of the required type.

4This is the fundamental example of a Kirchberg algebra, invented by Cuntz [16]. It is the universal C∗-algebra generated by n isometries whose range projections are orthogonal and add to 1. Cuntz proved that it is simple, and showed that On ⊗K is a crossed product of a UHF algebra (an inductive limit of matrix algebras) by an action of Z. But crossed products by Z preserve the category B, because of the arguments in Sections 2.4 and 2.5. Thus On lies in B. 24 JONATHAN ROSENBERG

Lecture 4. A fundamental example in noncommutative geometry: topology and geometry of the irrational rotation algebra

4.1. Basic facts about Aθ. We previously mentioned the algebra Aθ, defined to be the crossed product C(T) ⋊αθ Z, where T is the circle group (thought of as the unit circle in C) and where αθ sends the generator 1 ∈ Z to multiplication by e2πiθ, i.e., rotation of the circle by an angle of 2πθ. This makes sense for any θ ∈ R, but of course only the class of θ mod Z matters, so we might as well take θ ∈ [0, 1). This algebra has two standard names: a rotation algebra (with parameter θ), or irrational rotation algebra in the most important case of θ∈ / Q, or a noncommutative (2-)torus, because of the fact that when θ = 0, we get back simply C(T2), the continuous functions on the usual 2-torus. It is no exaggeration to say that these C∗-algebras are the most important examples in (C∗-algebraic) noncommutative geometry. In this section we’ll try to lay out the basic facts about these algebras, without attempting to prove everything or to explain the history of every result. The standard references for a lot of this material are the fundamental papers of Rieffel [48, 51]. A more extensive survey on this material can be found in [53]. We can describe the algebra Aθ quite concretely, using the definition of the crossed product in Section 2.2. The algebra has two unitary generators U and V , one of them generating C(T) and the other corresponding to the generator of 2πiθ Z. They satisfy the commutation relation UV = e VU. The algebra Aθ is the completion of the noncommutative polynomials in U and V . But because of the commutation relation, we can move all U’s to the left and all V ’s to the right in any noncommutative monomial in U and V , at the expense of a scalar factor of m n modulus 1. Thus Aθ is the completion of the polynomials m,n cm,nU V (with only finitely many non-zero coefficients). In fact, every element of A is represented P θ by a formal such infinite sum, but it is not so easy to describe the C∗-algebra norm in terms of the sequence of Fourier coefficients {cm,n}. The one thing we can say, since kUk = kV k = 1, is that the C∗-norm is bounded by the L1-norm, so that if the coefficients converge absolutely, then the corresponding infinite sum does represent an element of Aθ. (But the converse is false. This is classical when θ = 0, and amounts to the fact that there are continuous functions whose Fourier series do not converge absolutely.) The algebra Aθ has a canonical trace τ, i.e., a bounded linear functional with τ(ab)= τ(ba) for all a,b ∈ Aθ. We normalize by taking τ(1) = 1. Usually we add the condition that τ should send self-adjoint elements to real values, though when θ is irrational, this is automatic. When θ = 0, τ is just integration with respect to Haar measure on T2 (normalized to be a probability measure). There is a basic dichotomy between two cases. If θ is irrational, then no different 2πiθ powers of e coincide. It is not too hard to show from this that Aθ is simple and that there is a unique trace in this case, defined by the condition that τ(U mV n)=0 if m 6= 0 or n 6= 0. (Recall we do require τ(1) = 1.) So τ simply picks out the m n p (0, 0) coefficient c0,0 from m,n cm,nU V . On the other hand, if θ = q ∈ Q, then Aθ has a big center, and in fact Aθ is the algebra of sections of a bundle of 2 P ∼ matrix algebras over T . In fact one can show in this case that Aθ = EndT 2 (V ), the bundle endomorphisms of any complex line bundle V over T 2 with first ≡ p (mod q) (times the usual generator of H2(T 2, Z)). The algebra has many APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 25 traces in this case, but it’s still convenient to let τ be the one with τ(U mV n)=0 if m 6=0 or n 6= 0. (This along with the condition that τ(1) = 1 then determines τ uniquely.) The K-theory of Aθ can be computed from the Pimsner-Voiculescu sequence of Section 2.5. In fact, the main motivation of Pimsner and Voiculescu for developing this sequence was to compute K∗(Aθ). Since αθ is isotopic to the trivial action, regardless of the value of θ, the map 1 − α(1)∗ in (2.1) is always 0. Hence, just 2 as abelian groups, one always has K0(Aθ) =∼ K1(Aθ) =∼ Z . But one wants more than this; one wants a description of the generators. Tracing through the various maps involved shows that one summand in K0 is generated by the rank-one free module (or the projection 1), and that the two summands in K1 are generated by U and V , respectively. But the interesting feature is the order structure on K0, which comes from the inclusions of projective modules. Note that the trace gives a homomorphism from K0(Aθ) to R, sending a projective module to the trace of a self-adjoint projection (in some matrix algebra) representing it. (It’s a fact that every idempotent in a C∗-algebra is similar to a self-adjoint one; see for example [6, §4.6]. Since the trace takes real values on self-adjoint elements, the dimension of a projection is real-valued.)

Theorem 4.1. If θ 6∈ Q, the trace τ induces an isomorphism of K0(Aθ) with Z+θZ as ordered groups. If θ ∈ Q, then τ still sends K0(Aθ) to Z + θZ (which is equal to θZ in this case), but is no longer an isomorphism. The original proof of this theorem was nonconstructive, i.e., it did not exhibit a projective module of dimension θ that should be the missing generator of K0. We will talk about this issue later in Section 4.3. It follows from Theorem 4.1 that the irrational rotation algebras must split into uncountably many Morita equivalence classes, since it is easy to see that Morita equivalence preserves the ordering on K0, and since there are uncountably many order isomorphism classes of subgroups of R of the form Z + θZ. In fact, any order isomorphism Z + θZ → Z + θ′Z must be given by multiplication by some t 6= 0 in R, with the property that t ∈ Z + θ′Z and tθ ∈ Z + θ′Z. If we write t = cθ′ + d and tθ = aθ′ + b, a,b,c,d ∈ Z, then a b θ = · θ′ c d   for the usual action of 2 × 2 matrices by linear fractional transformations. Since the Morita equivalence must be invertible, we also have a b ∈ GL(2, Z). c d   So Morita equivalences of irrational rotation algebras correspond to the action of GL(2, Z) by linear fractional transformations. The converse is also true. Theorem 4.2 (Rieffel). Any unital C∗-algebra Morita equivalent to an irrational ′ rotation algebra Aθ is a matrix algebra over Aθ′ with θ in the orbit of θ for the action of GL(2, Z) on RP1 by linear fractional transformations. Every matrix in GL(2, Z) gives rise to such a Morita equivalence. This is not true by “general nonsense” but requires an explicit construction, which arises from the following theorem of Rieffel: 26 JONATHAN ROSENBERG

Theorem 4.3 (Rieffel [50]). If G is a locally compact group with closed subgroups H and K, then H ⋉ (G/K) and (H\G) ⋊ K are Morita equivalent. If we apply this with G = R, H = 2πZ, and K = 2πθZ, then H\G is the usual model of T and (H\G) ⋊ K is Aθ, while H ⋉ (G/K) is A1/θ. The Morita equivalence bimodule between these two algebras is a completion of S(R), with the two generators of each algebra acting by translation and by multiplication by an exponential, respectively. The reason why the two actions commute is that 2πis 1 translation by Z commutes with multiplication by e , while translation by θ Z commutes with multiplication by e2πiθs. The other Morita equivalences required by the theorem can be constructed sim- ilarly.

∞ 4.2. Basic facts about Aθ . One of the interesting things about Aθ is that it behaves in many ways like a smooth manifold. That means that we should have an ∞ analogue of the C functions inside the algebra of “continuous” functions Aθ. To 2 find this, note that Aθ carries an action of the compact Lie group T via (z, w)·U = zU, (z, w) · V = wV , for z, w ∈ T (viewed as complex numbers of modulus 1). This is analogous to the action of T2 on itself by translations. The smooth subalgebra ∞ ∞ Aθ is defined to be the set of C vectors for this action, i.e., the elements a ∞ 2 for which (z, w) 7→ (z, w) · a is C as a map T → Aθ. Alternatively, we can ∞ describe Aθ as the intersection of the domains of all polynomials in δ1 and δ2, the (unbounded) derivations obtained by differentiating the action. Since it is obvious that δ1(U)=2πiU and δ2(U) = 0, while δ2(V )=2πiV and δ1(V ) = 0, one readily ∞ sees (as in the smooth case) that Aθ is a subalgebra and that it can be described as ∞ m n Aθ = cm,nU V | cm,n is rapidly decreasing , (m,n ) X where “rapidly decreasing” means decreasing faster than the reciprocal of any pos- ∞ itive polynomial in m and n. Thus Aθ is isomorphic as a (not as an algebra) to S(Z2) and then by to C∞(T2). ∞ Proposition 4.4. The inclusion of Aθ into Aθ is “” (i.e., an element of the subalgebra is invertible in the subalgebra if and only if it has an inverse in ∞ the larger algebra), and thus the inclusion Aθ ֒→ Aθ induces an isomorphism on K-theory.

Proof. Isospectral inclusions preserve K0 and (topological) K1, by the “Karoubi density theorem,” so it is enough to prove the first statement. But this follows ∞ from the characterization of Aθ in terms of derivations, and the familiar identity −1 −1 −1 δj(a )= −a δj (a)a , iterated many times.  From this Proposition, as well as the fact that there is no essential difference between smooth and purely topological manifold topology in dimension 2, one might ∞ be tempted to guess that Aθ and Aθ behave similarly in all important respects. ∞ But a deep fact is that this is false; Aut(Aθ) and Aut(Aθ ) are quite different from one another. ∞ Theorem 4.5. If θ is irrational, every automorphism of Aθ is “orientation- 2 preserving,” i.e., the determinant of the induced map on K1(Aθ) =∼ Z is +1. On the other hand, Aθ has orientation-reversing automorphisms. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 27

Comment: The first part of this is due to [19]. The second part is due to Elliott and Evans [24, 23]. 4.3. Geometry of vector bundles. In classical topology, vector bundles play an important role in studying compact manifolds M. Recall Swan’s Theorem ([6, §1.7] or [20, §1.3.3]): there is an equivalence of categories between topological (respec- tively, smooth) vector bundles over M and finitely generated projective modules over C(M) (resp., C∞(M)). Thus in noncommutative geometry, finitely generated projective modules play the same role as vector bundles. Because of Proposition 4.4, when it comes to irrational rotation algebras, the “vector bundle” theory is essentially the same in both the continuous and C∞ cases, in that every finitely generated projective module over Aθ is extended from a finitely generated projective ∞ module over Aθ , which is unique up to isomorphism. Now K-theory gives the stable classification of vector bundles. The unstable classification is always more delicate, but for Aθ, this, too is known:

Theorem 4.6 (Rieffel [51]). For Aθ with θ irrational, complete cancellation holds ′ for finitely generated projective modules, i.e., if P ⊕ Q =∼ P ⊕ Q as Aθ-modules, for ′ ′ any finitely generated projective Aθ-modules P, P ,Q, then P and P are isomorphic. The isomorphism classes of projective submodules of a free Aθ-module of rank n are distinguished by the trace, and are given exactly by elements of K0(Aθ) =∼ Z + θZ between 0 and n (inclusive). Once one knows the classification of the “vector bundles,” in both the smooth and continuous categories, a natural next step is to study “geometry” on them. In his fundamental paper [14], explained how the theory of connections and in differential geometry can be carried over to the noncommutative case, at least when one has an algebra A like Aθ with an action of a Lie group G for which the “smooth subalgebra” A∞ is the set of C∞-vectors for the G-action on A. (This of course applies here with G = T2 acting as we described above.) Then if V is a finitely generated (right) A∞-module, a connection on V is a map ∇: V → V ⊗ g∗ (g the Lie algebra of G) satisfying the usual Leibniz rule ∞ ∇X (v · a)= ∇X (v) · a + v · (X · a), v ∈ V, a ∈ A , X ∈ g. Usually one requires a connection to be compatible with an inner product also. 2 ∗ Connections always exist and have a curvature 2-form Θ ∈ EndA(V )⊗ g defined as usual by V Θ(X, Y )= ∇X ∇Y − ∇Y ∇X − ∇[X,Y ]. ∞ Theorem 4.7 (Connes [14]). Every finitely generated projective module over Aθ admits a connection of constant curvature (i.e., with the curvature in i 2 g∗). The curvature can be taken to be 0 if and only if the module is free. More precisely, on the projective module with “dimension” p + qθ > 0, p, q ∈ Z, the constantV curvature connections have curvature 2πiq Θ(δ ,δ )= . 1 2 p + qθ Connes and Rieffel defined the notion of Yang-Mills energy of a connection, precisely analogous to the classical case for smooth vector bundles over manifolds. This is defined by YM(∇)= −τEnd(V ) {Θ∇, Θ∇} , where { , } is the natural on 2-forms.  28 JONATHAN ROSENBERG

Theorem 4.8 (Connes and Rieffel [15, 52]). If V is a finitely generated projective ∞ module over Aθ , a connection ∇ on V gives a minimum for YM if and only if it has constant curvature, and gives a critical point for YM if and only if it is a direct sum of constant curvature connections (i.e., V has a decomposition V1 ⊕··· Vn with respect to which ∇ has a similar decomposition into connections of constant curvature).

As we mentioned earlier, the original calculation of K0(Aθ) was nonconstructive, and the problem remained of explicitly exhibiting representatives for the the finitely generated projective modules. One answer is already implicit in what we have explained: if P is a finitely generated projective A-module, then it gives rise to a Morita equivalence between A and EndA(P ), so constructing all possible P ’s is equivalent to finding all Morita equivalence bimodules for A. In the caseof Aθ, they are all similar to the bimodule we mentioned before between Aθ and A1/θ. But one could ask for another answer to the problem, namely to give explicit representatives for all the equivalence classes of projections in Aθ (or in matrix algebras over it). Here two good solutions have been proposed, one by Rieffel [48] and one by Boca ∗ [7]. Rieffel constructed explicit projections in Aθ of the form Uf + g + fU , where f and g are functions of V . Boca instead constructed projections in terms of theta- functions which can be described as follows: if X is an A-B Morita equivalence bimodule as above, with A = Aθ, and if one can find an element ψ ∈ X with hψ,psiiB = 1B, then Ahψ,psii will be a projection in A. Boca’s projections come from choosing ψ closely related to a Gaussian in S(R).

4.4. Miscellaneous other facts about Aθ. Here we just mention a few other things about the algebras Aθ. The work of Elliott and Evans [24, 23], which we mentioned before, has more detailed implications for automorphisms and endomor- phisms of Aθ. Assuming θ is irrational, given any A ∈ GL(2, Z), there is an auto- 2 2 morphism of Aθ inducing the map A on K1(Aθ) =∼ Z , and given any B ∈ End(Z ) (including 0!), there is a unital endomorphism of Aθ inducing the identity on K0(Aθ) and the map B on K1(Aθ). Furthermore, the connected component of the identity in Aut(Aθ) is topologically simple, and Aut(Aθ) is just an extension of this con- nected group by GL(2, Z) [25]. All of this seems quite strange from the perspective of ordinary manifold topology, since a self-map T 2 → T 2 inducing the identity on K0(T 2) is of degree 1, and thus cannot induce the 0-map on K−1(T 2) =∼ H1(T 2). However, the endomorphisms constructed by Elliott’s procedure are unlikely to be smooth. Kodaka [38] did construct some special smooth proper unital endo- morphisms of irrational rotation algebras, but only when θ lies in a real quadratic number field. And one more structural fact about the algebras Aθ: they have real rank zero, that is, finite linear combinations of projections are dense in the set of self-adjoint elements. APPLICATIONS OF NONCOMMUTATIVE GEOMETRY 29

Lecture 5. Applications of the irrational rotation algebra in number theory and physics 5.1. Applications to number theory. 5.2. Applications to physics. 30 JONATHAN ROSENBERG

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Department of Mathematics, University of Maryland, College Park, MD 20742-4015 E-mail address: [email protected]